scholarly journals YOUNG MEASURES, CARTESIAN MAPS, AND POLYCONVEXITY

2010 ◽  
Vol 47 (2) ◽  
pp. 331-350 ◽  
Author(s):  
Patrick Bernard ◽  
Ugo Bessi
Keyword(s):  
Young Measures

On the computation of measure-valued solutions

Acta Numerica ◽  
2016 ◽  
Vol 25 ◽  
pp. 567-679 ◽  
Author(s):  
Ulrik S. Fjordholm ◽  
Siddhartha Mishra ◽  
Eitan Tadmor
Keyword(s):  
Fluid Dynamics ◽  
Materials Science ◽  
Numerical Algorithms ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Convincing Evidence ◽  
Young Measures ◽  
Inviscid Fluid ◽  
New Paradigm ◽  
Uniqueness Of Solutions

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

scholarly journals Nonlocal Variational Principles with Variable Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yongqiang Fu ◽  
Miaomiao Yang
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Variable Exponent ◽  
Variational Principles ◽  
Young Measures ◽  
Bounded Set ◽  
Weak Lower Semicontinuity ◽  
Variable Exponent Sobolev Spaces ◽  
Regular Open

This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.

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